We report on experimental measurements of how an externally imposed magnetic field affects plasma heating by kJ-class, nanosecond laser pulses. The experiments reported here took place in gas cells analogous to magnetized liner inertial fusion targets. We observed significant changes in laser propagation and energy deposition scale lengths when a 12T external magnetic field was imposed in the gas cell. We find evidence that the axial magnetic field reduces radial electron thermal transport, narrows the width of the heated plasma, and increases the axial plasma length. Reduced thermal conductivity increases radial thermal gradients. This enhances radial hydrodynamic expansion and subsequent thermal self-focusing. Our experiments and supporting 3D simulations in helium demonstrate that magnetization leads to higher thermal gradients, higher peak temperatures, more rapid blast wave development, and beam focusing with an applied field of 12T.
Recent interest in laser preheating for magneto-inertial fusion and Magnetized Liner Inertial Fusion (MagLIF) has encouraged experiments in magnetized laser-plasma heating in which laser-plasma interactions may be important.1,2 Laser-plasma interactions have been studied for decades, often in the context of inertial confinement fusion research. Among these are a number of experiments investigating reduced thermal transport in a magnetic field.3–5 However, there are fewer results from laser-plasma interactions in a magnetic field above a few Tesla which are strong enough to reduce electron thermal conduction. The physics of such interactions has found renewed interest because of magneto-inertial fusion and specifically Magnetized Liner Inertial Fusion (MagLIF) concepts.1,2 Here, we present experimental results and simulations describing how a magnetic field impacts length scales, axial and radial temperature profiles, and laser beam transport for high energy laser-heated plasmas by reducing heat conduction and enhancing thermal self-focusing (TSF).
An axial magnetic field will reduce radial electron heat transport in cylindrical laser-heated plasmas by approximately a factor of two.1,3 During heating, the thermal pressure exceeds magnetic pressure and hydrodynamic motion will not be directly affected by the field. However, if the Hall parameter χ⊥ (the product of the electron cyclotron frequency ωec and electron collision time τe) exceeds one, thermal transport will be reduced radially across field lines. In our experiments, we employ fields of 12–17 T such that the Hall parameter is in the range from ∼1 to about 10. This increases the local temperature and pressure gradients by reducing heat conduction. The coefficient of thermal conductivity κ for heat diffusion across magnetic fields is
where is the unmodified diffusion coefficient parallel to the magnetic field.6 The first expression is determined by analytically fitting a numerical solution to the Fokker-Planck relations for a Z = 2 helium plasma.7 The classical Bragginski transport form of Eq. (1) predicts χ⊥ = 5 in a plasma with kTe ∼ 300 eV with an imposed B-field of ≈12T at an electron density of ne ≈ 2 × 1020 cm−3, and thermal transport may be reduced by a factor of six. However, under the same conditions, the more exact formula will reduce transport by a factor of 12. Notably, even a Hall parameter of 0.2 is sufficient to reduce transport by a factor of three. Experiments with transverse Thomson scattering measurements by Froula et al. show that a field of 12T narrows laser-heated radial temperature profiles by 20%–30% and increases central electron temperature by more than a factor of two.3 The picture is complicated by the finite, varying radius of the laser and self-focusing. The dominant electron heat conduction modes and have different dependences and expressions when magnetic fields are moderately strong (χ⊥∼1) or very strong (χ⊥≫1). Nonlocal transport, magnetic advection, Biermann current, and other effects complicate heat transport modeling particularly in the intermediate field strength regime. For all these reasons, any data we provide on temperature profiles, heated widths, and axial heating depth will be valuable for understanding heat transport and energy deposition in magnetized laser-heated plasmas. The data most relevant to this problem have been recently published by Carpenter et al. who measured peak plasma temperature in deuterium gas cells seeded with a small amount of Ar under conditions relevant to MagLIF with and without an axial magnetic field.8 They found that higher peak temperatures were achieved when an 8.5 T magnetic field was present, and they observed that the laser heated plasma columns were longer in the presence of that B-field. These measurements, however, were not time resolved and were not 2D spatially resolved, so they could not discern laser propagation effects in the magnetic field, such as self-focusing.
Laser propagation through plasmas is complex, including a number of laser-plasma instabilities (LPIs) that have been well studied in literature. These are less well studied for magnetized laser plasmas, for example there are few experimental results on magnetized self-focusing LPIs. We experimentally investigate LPI regimes in which TW-class, kJ lasers heat plasmas within magnetic fields exceeding 10T using the Z-Beamlet laser at 3.5 ns, 1/3 TW, heating ∼ncrit/20 plasmas, where ncrit is the critical density. Electromagnetic propagation in a magnetized plasma differs, and the dispersion relation becomes that of Eq. (7) in Wilson et al.9 In our case (<), the dispersion relation will not significantly differ from that of a laser in an unmagnetized plasma, and laser propagation will proceed unaffected by modified cyclotron gyration. Some LPIs are not expected in this regime; the two-stream instability of counterstreaming ions and electrons is not expected in our experiments.10 Similarly, relativistic and ponderomotive self-focusing are not expected above the pondermotive critical power, which is orders of magnitude above our laser power.11,12 However, we expect TSF with and without a magnetic field, causing both whole-beam convergence and <0.1 mm scale self-guided filaments.
Simulations and theory indicate that a magnetic field should enhance TSF and laser filamentation,13 but this is not well studied experimentally. A magnetic field will increase local temperature gradients and thus the field intensifies hydrodynamic expansion since the magnetic field pressure is far less than the plasma pressure. Consequently, a magnetic field co-aligned with the laser axis is expected to enhance TSF. TSF occurs at different lengths scales, resulting in either small-scale filaments or symmetric whole-beam self-focusing. The parameter space we explored experimentally is similar to Watkins and Kingham's simulated and analytical calculations for whole-beam self-focusing and smaller-scale filamentation in densities, fields, and laser intensity. In this paper, we observe results that bound their predictions.
The underdense plasmas in our experiments are principally heated by inverse bremsstrahlung (IB) absorption, creating a well-understood temperature profile Te(r = 0,z,t) described as a bleaching wave. The IB absorption coefficient κIB scales with temperature as kTe−1.5. As the laser heats a region to higher temperature, the absorption and opacity decrease, allowing the heating to penetrate more deeply into the plasma. The laser intensity I along propagation through the plasma is given by dI/dz = −κIBI. The solutions for I(z) and Te(z) were derived by Denavit and Phillion14 and independently by Slutz.1 The solution for the axial variation of the temperature was found to scale such as
where Te,0(t) is the temperature at z = 0 which is continually increasing in time, Φ0(t) is the intensity of the laser pulse in time, and zf is the front location. Thermal self-focusing and magnetization alter this axial heating by reducing the area and radial conductivity of the heated region, respectively. Magnetization will increase laser-heated central Te and thus reduce IB absorption, with the overall effect of lengthening the absorption length. This can happen concurrently with TSF, which narrows the heated cross section. As we will describe in the results section, we used x-ray diagnostics to image plasma emission and length scales. The combination of TSF and magnetized IB heating makes it difficult to distinguish between exact causes for length scale and radial profile changes, which suggests that detailed experimental work is required.
Our experiments aim to emulate laser pre-heating in magneto-inertial fusion (MIF) schemes, e.g., MagLIF.1,2 Our experimental conditions were like those used in preheating of early MagLIF experiments; they differed mainly in the choice of helium as the target gas instead of deuterium, a gas vessel with length and radius that are more than three times larger, and laser focusing with an f/6.6 optic instead of an f/10 lens.1,2 Our applied magnetic fields were in the same range as initial MagLIF experiments, which used 10 T.2 TSF may be detrimental to MagLIF preheating, because it decreases laser coupling efficiency via parametric LPIs particularly in high intensity focused areas.6,14 These LPI processes include stimulated Raman and Brillouin scattering (SRS and SBS) which create side-scatter and backreflection and which other work is attempting to both measure and minimize.13,15,16 LPI effects create laser absorption inefficiencies and are mitigated in our experiments by reducing the beam power from 1 TW to less than 500 GW and by the use of 1 mm distributed phase plate to increase the focus size.15 These measures mitigate TSF, which reduces the overall energy coupling efficiency because higher laser intensities increase temperature and reduce IB coupling. TSF also increases density gradients and decreases density in the beam through hydrodynamic expansion. Furthermore, the beam can filament into hot spots, leading to increased backscatter or sideways spray. TSF is expected to scale adversely with heating time, central temperatures and temperature gradients, and overall coupled energy. We present the first experimental examination of the enhancement of TSF with a magnetic field using the Z-Beamlet laser in a setup tuned to MagLIF parameters.
II. EXPERIMENTAL SETUP
We developed a set of experiments with laser heating embedded in an externally imposed magnetic field at Sandia's Z-Beamlet Laser facility (ZBL).17 Laser heating of a pre-magnetized gas occurred inside a gas cell target with pulsed axial magnetic field Bz ranging from 5 T to 17 T as shown in Fig. 1. Fields of 12 T were used in the shots discussed in this paper. The Z-Beamlet laser created plasmas with pulses of 1–1.5 kJ energy, 527 nm wavelength, and 2–5 ns pulse duration, yielding ∼1/3 terawatt peak power.18 The gas cell contained helium with neon dopant, with overall electron density ne ∼ 2 × 1020 cm−3, or 5.5% ncrit. We fielded diagnostics to characterize energy density and temperature transport in the plasma, as well as the heating characteristics, both radially and axially. The primary diagnostic was an x-ray pinhole camera (PHC) with a time-gated 2D image sensor framing the full extent of the plasma. A secondary diagnostic was a transverse shadowgram imager time-gated with a 100 ps resolution capturing shadowgraphs of the backlit plasma. A pulsed 532 nm laser relay-imaged onto the imaging detector was used for these shadowgraphy measurements. The x-ray images were captured along a viewing axis that was 90 degrees from the shadowgrams.
Our targets were plastic gas cells with a 1.9 cm inner diameter. These cells had a 2 μm polyimide laser entry window (LEW) for the ZBL heating beam that also served the purpose of containing the target gas. We diagnosed the generated x-rays through a transverse 2 μm thick polyimide window that was 10 × 3 mm in size. Shadowgraphs were generated using probe lasers through transverse visible light viewports. The gas cells were surrounded by a copper coil with two parallel turns that spanned the diagnostic ports.17 The magnetic field strength generated by the coils on axis dropped by approximately 30% between the loops as these were farther apart than a nominal Helmholtz coil configuration. Magnetic fields imposed in the data presented here were 12 T between the coils at z = 6 mm generated by 400 kA currents with 2 μs rise time. Aside from the coils, the gas cells used no conductive materials, allowing the applied B field to magnetize the target interior.
We operated ZBL with energies ranging from 1 to 1.5 kJ with 3.5 ns constant-power (flat-top in time) heating pulses at 0.3–0.4 TW, delivering a maximum of about 1.3 kJ to the target after losses on transport and focusing optics. The beam waveform and a 1D radial intensity distribution are given in Fig. 1. We used a distributed phase plate (DPP) designed to yield uniform irradiance to within 5% over a 1.5 mm circle when used in conjunction with a 3.2 m focal length, f/10 lens. In our experiments, this DPP was actually used in combination with a slightly shorter, 2 m focal length lens, so it produced a heating beam spot of 0.95 mm diameter. This configuration generated a pedestal irradiance of mid 1013 W/cm2 with the 0.3 TW pulses. The heating beam temporal waveform intentionally included a 30 J prepulse 20 ns before the main 3.5 ns long pulse to ablate the laser entrance window to a density well below ncrit.19–21
A. X-ray pinhole camera diagnostic
We fielded pinhole camera (PHC) detectors looking transverse to the laser-heating axis to capture plasma x-ray emission from neon line radiation and bremsstrahlung. The PHC detectors use Sandia-developed hybrid-CMOS Icarus sensors with 2 ns exposure times and 2 ns inter-frame times.22 The 25 μm pixels are smaller than the pinhole size and do not limit the resolution in the x-ray images. These sensors acquire up to four frames on the same sensor surface. Sub-dividing the sensor with custom time delays and arraying pinholes in parallel enabled up to 16 exposures of the evolution and transport of the full extent of the plasma per shot. The detector energy sensitivity calibrated with a 1.35 keV monochromatic photon source was linear with incident photon counts in the range from 0 to 2500 photons, with a pixel calibration of 2.9 keV/count.
The target gas was helium at four atmospheres. Neon was introduced into the He at 0.1% by atomic fraction to serve as a 900 eV K-shell x-ray emitting dopant. X-rays were attenuated by the 2 μm of polyimide gas cell window and a 6 μm aluminum filter. The aluminum filter largely passed x-rays below the Al K-shell edge in the range of 0.8–1.5 keV. Figure 2 shows the full spectrum of filtered x-rays for this gas and filter choice, with emission spectra calculated using the PrismSPECT code23 and filtering data from Henke.24
Figure 3 shows the calculated integrated filtered x-ray radiated power that reaches the detector as a function of electron temperature. The scaling is nearly linear in the temperature range of 100–380 eV. Beyond this electron temperature range, neon K-shell emission saturates and the bremsstrahlung continuum contribution from helium dominates although bremsstrahlung does not significantly increase emitted power up to 1 keV in plasma temperature. The x-ray emission expected is also very low below electron temperature of 125 eV, so it does not allow us to discern cooler regions of plasma like the radiative precursor plasmas we expect around the main heated column. Consequently, this diagnostic was a good measure of the region of principal laser energy deposition although it is not well suited for measuring temperatures when they are elevated above 400 eV as they are in some regions of the plasmas we produce. At temperature between 100 and 380 eV, the x-ray power is proportional to electron thermal energy density, and pressure, giving us a diagnostic for exploring temperature and pressure distributions both radially and axially in this region of energy density.
Here, we present data from two representative shots, one with a 12 T applied magnetic field and one without, from a shot series on the Z-Beamlet laser. These shots demonstrated changes in plasma length scales, temperature trends, hydrodynamic motion, and filamentation with an applied magnetic field. Figure 4 shows an x-ray pinhole camera image sequence of these two sample shots with the following setup parameters: (1) both the inter-frame delay between images and the integration time are about 2 ns. Each successive image in row B is delayed by 2 ns from row A in the image sequence. (2) Matching pinholes pairs are used and are 100 μm on the left and 200 μm on the right. (3) The gas cell is filled with 4 atm He, 5 Torr Ne (0.16% atomic fraction). (4) A 3.5 ns heating pulse is used in both cases, with about 1.25 kJ delivered to the 12 T shot and 1.39 kJ delivered to the non-field shot. (5) Each shot created a 1 mm diameter DPP-conditioned laser spot on the laser entry window.
The finite integration time of 2 ns introduced an averaging uncertainty and image blurring. Every interpreted temperature that we present in this paper is calculated using an averaged power over a 2 ns interval. Spatial information has motion blur from hydrodynamic and thermal expansion, and outer plasma edges are interpreted as describing the end of the relevant frame interval.
A. Magnetic field effects in heated plasma length scales
1. Axial lengthening with a magnetic field
Analysis of time-gated x-ray PHC image sequences provided information on the evolution of the plasma dimensions. We estimated the plasma radius at each axial location along the plasma column by finding the radius at which the x-ray emission intensity dropped to 5% of the radial maximum. Axial cutoffs locate the start and end of the plasma in the image with a 5% of overall maximum cutoff in the background-subtracted average of emission. Figure 5 shows that the most strongly emitting plasma regions increased in axial length with a 12 T applied-magnetic field by 15%. Visible shadowgraph images (see Fig. 7 below) underlaid with x-ray data indicate that an ionizing radiative precursor is also longer by about 1–2 mm or 15% in the case in which a B-field is applied. An identical pair of shots not presented here as well as several similar shots with slightly different initial parameters were performed with similar outcomes.
Axial lengthening will take place if reduced transport creates a hotter channel with lower IB absorption. This effect alone can explain heated length changes if radial heat transport is reduced by a magnetic field. Reduced radial transport causes higher central temperatures and lower central densities, leading to lower IB coupling and greater penetration. 1D models of IB heating with no radial heat transport (dotted and solid curves in Fig. 9) and subsequent 2D and 3D HYDRA simulations displayed this effect. Simulations indicate that a magnetic field's impact on IB absorption lengthens the plasma axially by 2 mm.
We also expect axial lengthening from self-focused LPIs even in the absence of a magnetic field. This will occur via whole-beam or high energy self-focused filaments. With our modest intensity compared to relativistic laser field scales, and with high densities and nanosecond pulses employed in our experiments, TSF dominates over ponderomotive self-focusing.25 We can see this by examining the thresholds for pondermotive and relativistic focusing, for example, using Eqs. (7)–(9) from Williams.26 Since the electron relativistic speed in the laser field is a factor of 100 smaller than the thermal speed, resulting ponderomotive density variations will be of order . By contrast, our few nanosecond, 527 nm kJ-class heating pulse can create keV temperatures, pressure gradients of 0.1 Mbar/mm, and >50 km/s sound speed hydrodynamic motion. TSF results from rapid radial hydrodynamic motion and cavitation in regions of the highest temperature during laser heating. The alteration to the plasma density profile changes beam transport properties by altering the radial refractive index profile.27–29 This creates a refractive gradient, focusing the heating pulse inward. Density depletions >40% are expected after a few nanoseconds, which are much greater than pondermotive density variations. TSF self-focusing length for Gaussian beams is on the centimeter scale based on approximations from equation Estabrook et al.26 TSF will be enhanced by a magnetic field.
Whole-beam and filamentation TSF can also narrow the heating beam further and subsequently cause deeper heating penetration. Our experiments cannot conclusively determine if TSF is a contributing effect to the overall length increase with an applied field. Whole-beam TSF causes a beam convergence to a hot spot and can subsequently enhance LPI's like SBS or ponderomotive focusing which require higher intensities of order . 3D HYDRA simulations in Fig. 11 predict an axial peak in temperature downstream in the magnetized case spanning z = 3–5 mm. This situation is discussed in Watkins and Kingham's paper, for example, in Fig. 7, a 10 T magnetized plasma with density and laser irradiance within a factor of two of ours achieves TSF convergence at z = 10 mm.13 Bright spots in x-ray emission are present past z = 7 mm in downstream in x-ray images of magnetized shots (Fig. 4) in frames captured in a time window 3–5 ns after heating. Since x-ray emission scales linearly with density and hydro motion takes 1–2 ns, this downstream emission peak is best modeled by the regions of >400 eV temperature which have not begun significant density depletion. The full-width half-maximum of the radial distributions of these downstream x-ray bright spots are still only 0.7 times the original heating beam width. A sub-millimeter downstream heated width can also be explained by radial gradation in IB opacity, but the increased emission requires very high temperatures into the regime above 1 keV where helium bremsstrahlung emission increases significantly. In general, the change in axial heating depth from magnetized TSF is difficult to quantify experimentally because TSF can simultaneously lengthen the heated area by narrowing the beam, cause filamentation that disperses energy, and halt primary heating beam propagation by converging the beam to a hot spot.
2. Radial narrowing from reduced transport
We observe radial narrowing of the x-ray emission region at early times with the 12 T magnetic field, as shown in Fig. 5, consistent with reduced electron thermal diffusion across the magnetic field. Reduced radial electron heat transport will inhibit heat from escaping radially, which reduces the width of the x-ray emitting region from what it would be otherwise. In the temporal snapshots of these images in the upstream regions closer to the entrance window, thermal transport and hydrodynamic motion have had more time to occur.
X-ray emission following laser heating diminishes rapidly, as seen in Fig. 4, as the plasma cools from radiation and radial thermal heat conduction. We attempted to quantify the differences in the radial emission profiles with and without B-fields by finding the radial extent of the emission at each axial location. We did this by taking a weighted average of the radial emission intensity W(r). We calculated ∑W r/∑W at each radial lineout along the plasma column, where r is the radius from the distribution's centroid. This analysis is shown in Fig. 6. These data demonstrate that the plasma in the B = 0 case is consistently wider by about 10% throughout the axial length of the plasma mid-way through the laser heating process (a widening of observed heated region by about 100 μm). This difference in plasma width is also consistent with 3D simulations presented in Sec. VI. Reduced radial heat transport produces a narrower heated region around the beam when a B-field is applied.
Around the end of the laser heating, in the 3–5 ns time window, the differences in the x-ray emitting radius between the B-field case and the non-field case decrease with z as shown in Fig. 5. The B = 0 case is wider in the upstream region because radial diffusive transport is active for s longer time at the time the plasma image is recorded (because the laser pulse passed that upstream, axial region earlier in time).
In the B-field case, the narrowest profile and highest-temperature x-rays are emitted downstream.
B. Filamentation and TSF enhancement with an applied magnetic field
We sought to validate that both whole-beam and filamented TSF is occurring and enhanced by a magnetic field. Our x-ray imaging used 2 ns time-average x-ray imaging and 100–200 μm, so it is sensitive to whole-beam self-focusing and overall trends in heating. TSF can cause increased axial length, a self-focused hot-spot, and filaments diverging from the plasma in either case. Reduced radial transport alone would explain the increase in axial length via IB heating opacity reduction. However, the presence of a bright spot in x-ray emission downstream in Fig. 4 in a downstream location indicates whole-beam self-focusing with applied-field. This is also evident in 3D HYDRA simulations (Sec. VI) in Fig. 11 and not present without an applied field.
Simulations predict filamentation in both a magnetized and unmagnetized case as we see in Fig. 12 from HYDRA. There is some evidence in visible light that a magnetic field enhanced downstream filamentation. In the visible light shadowgrams of shots with an applied 12 T magnetic field, we observe parallel axial features in the downstream end of the plasma suggesting filamentation; these are absent in shots without a field. Figure 7 shows shadowgrams captured within 300 ps of the end of the 3.5 ns heating Z-Beamlet pulse. Note that shadowgrams show a very sharp-edged plasma boundary corresponding to the edge of a low-temperature ionized precursor, outside of overlaid, hotter, x-ray emitting plasma images. These edges are created because emissive helium plasmas with Te > 25 eV create a pedestal-temperature radiation-diffusive precursor with ∼25 eV UV photons in thermal equilibrium with a plasma near the ionization energy.30 This precursor occludes visible light at a larger radius than the region emitting Ne K-shell x-rays in the main heated plasma column and is several hundred micrometers wider. The filament structures in images are approximately 1 mm wide and emerge outside of the ionized precursor edge. 3D HYDRA simulations, discussed in Sec. VI, show filamentation in both cases. However, with a magnetic field, more energy enters filamentation and filament propagation length goes from ∼3 to 4–5 mm after 3 ns of heating. In simulations, filaments form most readily downstream and most evidently after nanoseconds of heating. In simulations of the magnetized case, the beam also exhibits whole-beam TSF in the plasma column, redirecting rays toward the axis. Rays thus form a hot spot and subsequently spray into an array of forking filaments which extend forward several millimeters. The filament features we imaged in shadowgrams with magnetized heating are consistent with enhanced filamentation in simulations although the forking is less pronounced and the imaged filament features are highly axisymmetric.
X-ray images do not show downstream filaments despite simulations predicting their appearance in x-ray frames, for example, in Fig. 14. Simulations generated 3D filament features that rapidly evolve in time, and the x-ray images are smoothed in time and space; this may be a factor. Possible explanations for the lack of downstream filaments visible in x-ray emission may include unaccounted for thermal conduction or MHD effects like azimuthal current or magnetic advection. The discrepancy between experiments and simulations in this way warrants further study.
In experiments, we observe an x-ray emission bright spot around z = 7–8 mm in the applied B-field case that is absent without the field. This bright spot is notably not axisymmetric in one of our shots shown in Fig. 7. In other shots, the full-width half-max of this bright region is narrower by about 45% than a 1 mm diameter best-focus diverging laser pulse at f/6.6 beyond the ∼5 mm Rayleigh range. Watkins and Kingham observed similar effects in their simulations where an applied B-field will decrease that axial depth at which filamentation is expected, as well as enhancing whole-beam TSF, which can cumulatively reduce whole-beam propagation to less than three millimeters length in some cases.13 Downstream heated areas have had less time for pressure-driven density depletion and will emit more strongly. The central density depletion which causes an emissivity drop on-axis is more significant in the applied field case as density and temperature fits from data analysis indicate. Overall emission in these downstream bright spots is perhaps 15% narrower by FWHM than emission from heating upstream on earlier frames, so additional beam is convergence potentially present there compared to upstream, but other effects may explain this without TSF, such as radially increasing IB coupling. By comparison with the magnetized case, the B = 0 case displays axially uniform emission throughout all early frames, which suggests neither a TSF focal hot spot nor enhanced hydro motion. The emission bright spot in x-rays and visible filaments are consistent with increased density gradients in the applied magnetic field case, driving TSF. The experiments are consistent with the beam converging downstream and then diverging into small filaments and breaking up as it does in simulations, but our experiments cannot conclusively say this.
IV. INTERPRETATION OF RESULTS
A. X-ray image analysis
Our x-ray diagnostic is not ideally suited to a quantitative measurement of the electron temperature because of the flattening of the Ne emission when electron temperature is above 400 eV. The plateau in emission above this temperature precludes measuring temperature and overall deposited energy above a certain threshold using x-ray emission. Nonetheless, because we observed clear qualitative differences in the radial profile of Ne emission between the cases with and without magnetic field, we did attempt to derive further information by fitting the emission. To this purpose, we model emissivity using cylindrically symmetric radial temperature Te(r) and density ne(r) distributions that are self-similar thermal waves and blast wave density approximations informed by simulations. These distributions are given in Sec. IV B in Eqs. (4) and (6). We calculated the expected x-ray emitted power as a function of plasma temperature at each point using an interpolation lookup table P(Te) plotted in Fig. 3. We created this table using the PrismSPECT code23 to give the spectral emissivity (Fig. 2), and we integrated this over the spectrum of our helium-neon mixture at various temperatures, with appropriate filters. We then integrated the analytical emission profiles via forward Abel transform along the line of sight of each 25 μm pixel through the 1:1 pinhole camera. We assumed that the plasma is optically thin to x-rays and found the projection onto the two-dimensional observation plane of the detector. Using the image width and maximum emission to constrain the result, we automated the generation of an inverse function in our analytical fits to define free parameters Te(r) and ne(r) that reproduced the measured x-ray radial emissivity profile at each pixel. Generally, we used types of Te(r) and ne(r) solutions to (1) constrain a thermal wave model with a temperature scaling in the thermal conductivity χ of 5/2 and assuming the central density dropped by about 40% to form a blast wave on the boundary (estimated from HYDRA simulations discussed below) and (2) constrain a thermal wave with a parabolic profile with a central density that drops and a blast wave with four times the density (the compression expected on the Hugoniot of an ideal gas). We used case (2) for a centrally hollowed measured profile with an applied magnetic field. Simulation results in subsequent sections offer comparisons to such models. This analysis was performed for each frame at each axial position.
We accounted for a pinhole transfer function using the solid angle fraction of the pinhole. We used a Gaussian pinhole transfer function, approximating an Airy function, to estimate the spatial blurring the pinhole created. We included x-ray filtering from a 2 μm thick plastic window and 6 μm aluminum filter. There is a large uncertainty in the temperatures we calculate from data for the magnetic field cases because these plasmas were hotter, and as Fig. 3 illustrates, the diagnostic cannot distinguish temperatures higher than 400 eV. Simulations indicate that the temperatures with a magnetic field present may be as much as a factor of two higher, approaching 0.9 keV.
B. Electron thermal wave radial profiles
Our analysis of x-ray images required radial emissivity models with analytical approximations for radial temperature and density distributions, which were informed by simulations similar to Fig. 13. We used primarily a self-similar thermal wave as an analytic radial temperature profile estimate. Zel'dovich and Raizer solved for thermal waves with diffusion that scales as an exponent of temperature.30 In cylindrical geometry, the solution is
where Tc is the axial peak temperature, rf(t) is the heat front radius, and y is the power scaling of temperature in the thermal conductivity. The well-known Spitzer–Harm (SH) conductivity has κ ∝T5/2 for electrons.31 This should work fairly well for the unmagnetized case as we are likely to be able to apply the fluid equations given a Knudsen number of 10−4–10−1 up to 600 eV. While a Knudsen number of 0.01–0.1 permits some nonlocal behavior at higher temperatures, the continuum heat flow is applicable to most regions except in regions of laser hot spots from TSP. Diffusive behavior still applies to the bulk of the plasma and the periphery where a heat front expands. The time dependence of Tc(t) and rf(t) is solvable for a thin axial line heat impulse initial condition where Tc decreases with time as the wave propagates. The solution for the heat front radius, using methods from Zel'dovich and Raizer, is as follows:
where “a” is the coefficient of the temperature dependence in the thermal conductivity κ used in the heat equation and Q is a real integrated heat in a thin differential disk in units of temperature times area. Q is related to the total thermal energy per unit of length in the plasma column via E ∝ nQ where n is number density.
The solution in Eq. (5) does not apply at early times in the laser pulse because the laser energy is not a spatial delta function but is extended over the laser spot size, and the laser heating occurs over finite pulse duration. It is a reasonable approximation, however, as these driven heat waves evolve rapidly into self-similar spatial profiles as heating continues.32 For this reason, it is still a good estimate to treat the thermal profile as a self-similar solution. Using this approximation, we determine rf by examining the radial x-ray data, and then, we calculate Tc. For example, in the left panels of Fig. 8 with time window of 1–3 ns, we compare radial x-ray data with a fit to the data generated from such a Te(r) in Eq. (4) with y = 5/2.
In general, without an applied B-field, a thermal wave solution using a Spitzer–Härm (SH) thermal conductivity with y = 5/2 fits our data for early times without any density modification. This implies that little hydrodynamic motion has occurred, and the temperature profile is merely that of the thermal wave propagating in uniform density plasma. In these conditions, above around Te = 160 eV, the Hall parameter χ⊥ exceeds 1. For such conditions, Eq. (1) shows κ⊥ decreases with a temperature exponent of y=−1/2 as thermal conduction of hot electrons is suppressed. With such a scaling, a self-similar heat wave solution is no longer permitted. However, to analyze radial data with a magnetic field present, we chose y ∼1 rather than y = 5/2 in Eq. (4), producing an inverted parabolic temperature profile. This profile is further informed by simulation results, such as those in Fig. 13. Such a choice may be justified if, in fact, radial cross field transport did not decrease with temperature following classical transport but instead scaled linearly with temperature, neglecting other effects like radiation conduction. Other effects are known to potentially create radial variations in both the density and magnetic field including hydrodynamic motion and Nernst advection. Such effects will modify the collision frequency and cyclotron frequency in a radially dependent way, so that any analytic heat flow solution would then need a nonlinear κ⊥(r). To best capture this requires a full 2D or 3D simulation, which we present later. Regardless, an inverted parabolic temperature profile with y ∼1 is a good fit for experimental results from emission profiles while also being a close analytical approximation of simulations.
Analytically fitting radial emissivity trends in data on 2+ ns heating timescales required modeling hydrodynamic motion which modified emissivity. The thermal wave solutions alone cannot fit our data well later in time and for the 12 T magnetic field case on the right in Fig. 8. This was also the case to a lesser extent without an applied field. Prior to 1 ns of laser heating time, steeper profiles without density modifications fit downstream, while upstream the emission profiles become flatter where blast waves have had ∼2 or more nanoseconds to develop. To quantify this and determine if the profiles we observe are consistent with enhanced hydrodynamic expansion, we fit our data with a radial density that closely approximates a blast wave. We used a combination of analytic approximations for a radial thermal wave radial temperature profile and a blast wave density profile to fit the data. X-ray emission scales linearly with ne(r) and increases with temperature as given by Fig. 3. As an analytic approximation, electron thermal energy density can take the form
In Eq. (6), the first two terms of the right-hand side contain the temperature dependence in a thermal wave as in Eq. (4). We used y = 1 with applied B-field and y = 5/2 for a no-field case to reproduce data in Fig. 8, following analytical theories and informed by temperature profiles like those in Fig. 13. is the background gas electron density for a fully ionized plasma, and is the density at r = 0. The rightmost term is an analytic approximation of electron density that results when a blast wave forms on the plasma periphery based on density profiles from Fig. 13. Constants in the rightmost term ensure particle conservation. In our analysis, we did not use Eq. (6) for emissivity directly, but instead modeled emissivity that scales linearly with density while using the temperature scaling in Fig. 3. With this fit for , the combination of a central peak temperature and centrally hollowed density with strong blast waves will produce orthogonally projected emission that is radially uniform, similar to the flat emission profile data in Fig. 8. Doing this yielded emission fits like those in Fig. 8 in the B-field case at z =4 mm. The drop in central density was also scaled linearly in z such that very little density depletion occurred downstream at the time of heating, which also reproduced the data very closely.
The best fits for x-ray emission with a magnetic field have steeper thermal gradients (Grad T a factor of two higher) and greater hydro expansion (2× higher blast wave peak density after 4 ns). These results agree with HYDRA simulations, which have steeper thermal gradients and subsequently enhanced hydro motion in the applied field case. Our results validate a reduced thermal conductivity and reduced thermal conductivity. They also demonstrate more dramatic blast waves with an applied field. This can be seen in the frames captured 4 ns after the start of heating and 2 mm from the window, best fit central density in the unmagnetized case and in the magnetized case. These enhanced density gradients which will more strongly focus the tail end of the laser pulse via the TSF mechanism. This is the most compelling evidence that the magnetic field enhances density gradients: annular emissivity profiles can only be generated consistently by annular blast wave density profiles. In addition, these early time blast waves are still developing, and their finite initial radius and accumulating density prevent extracting energy density data from their trajectory, as later frames have done.19
V. GENERAL AXIAL TRENDS IN TEMPERATURE AND ENERGY
A. Basic models for axial temperature profiles
To understand axial trends in temperature and deposition length, we developed a simple 1D model to calculate the effects of heat transport and collisional absorption. As laser energy is deposited with IB, the local temperature is updated each time step from increased energy. We calculated temperature from the energy in each plasma disk via E = neVkT + niVI where V is the disk volume, and ni and ne are the respective densities. I is the total ionization energy per ion as a function of the average ionization state Z based on temperature in the previous time step. I and Z were built from a linear interpolation from average ionization data as a function of temperature from a collisional radiative ionization model, FLYCHK.33 With this we calculated an expected central electron temperature. Calculations of the peak temperature on axis as a function of laser propagation length under various transport and focusing conditions are plotted in Fig. 9. The equations from Denavit and Phillion are shown as the solid curve in Fig. 9.14 Other lines reproduce estimates from our simple model. In all cases, we assume a 3.7 ns heating pulse duration and an initial heating radius of 0.5 mm. The target gas is pure helium at 2 × 1020 cm−3. The simplest 1D model uses a constant heated diameter and matches closely with the analytical solution. We created another basic 1D model that included free-streaming radial heat transport at a speed limited by the plasma sound speed on the order of 0.1 mm/ns. This expanding thermal radius model assumes that the hot region has a sharp boundary expanding with r(t) at constant velocity into cold gas, conserving energy and reducing local temperature. This result produced a significantly shorter axial length of heating and lower overall temperature plasma, and represents an upper limit on radial transport without a magnetic field.
We wanted to explore the axial temperature trends that would result from a decreasing heated radius (which might occur with TSF). We made rough estimates by modeling a heated region that shrinks in size as it propagates axially linearly with z. We used this conically converging heating beam model to validate our understanding of a heated region with time-dependent thermal expansion in the absence of a magnetic field and set up an adequate comparison with magnetized cases. This simple approximation assumed a beam propagating into the plasma with a diameter of 1 mm which converged to a constant smaller diameter at z = 0.4 mm and then a constant beam diameter beyond that point. The converging beam model in Fig. 10 uses an axially decreasing heated radius from 0.5 to 0.3 mm radius over 4 mm. The transition results in the inflection point in the converging curve in Fig. 9. The temperatures we predict in Fig. 9 are high, above 1 keV in much of the heated column. The range of length of the heated column predicted is consistent with observations and highlights the effect of changes in the heated area and the way that radial heat transport lengthens the plasma and radially narrows the heated regions.
B. Self-focusing increases with density
Guided by this simple model, we can also understand beam transport and heating dynamics by analyzing trends in heated axial length as a function of gas fill density. We used the visible light shadowgraphy diagnostic on a larger collection of shots to measure plasma length from the laser entry window at various helium densities without an applied magnetic field. These shots provide a baseline comparison of energy deposition length in the unmagnetized case, as shown in Fig. 10. The shots occurred with a variety of heating beam setup parameters and input energies. A monotonic, approximately linear decrease in heated length with increasing density is evident.
To understand this trend, we compared data with axial lengths predicted from the simple 1D numerical IB heating model, discussed above. They were run at various gas fill densities, and the heated length was extracted for each case. There are three variations to the model, starting with a basic model in which heating area stays constant in z. The length as a function of density is estimated for a diverging heated area (with the f/# of the focusing lens) is shown as a line with short dashes in Fig. 10, and the estimate assuming a converging heating area is shown with longer dashes. In all three calculations, the length of heated plasma scales approximately like ne − 3/2. Overall, experimentally the plasmas tend to be longer at high density than this basic 1D model, but the trends in energy deposition length seen in the various data seem to be reasonably reproduced by the converging heated area. Radial heat transport reduces upstream temperature and increases initial heated area, and therefore enhances local energy absorption, so it will shorten the plasma rather than lengthening it.
VI. SIMULATION RESULTS
We performed 3D HYDRA34 radiation-magnetohydrodynamics simulations of these experiments with and without an applied magnetic field. The HYDRA simulations used a ray trace model of the Z-Beamlet power time-history, DPP spot profile, and beam cone angles. The simulations include modeling of inverse bremsstrahlung heating ponderomotive effects (pressure and momentum transfer), ablation dynamics, and mix of the target entrance window but do not include modeling of LPI processes such as stimulated Raman and Brillouin scattering. While the laser package is well converged, we do not resolve the speckles that occur in the DPP spot which is impractical for these calculations. Additionally, HYDRA lacks important physics, such as diffraction, that occurs at those scales and would require a more appropriate laser-plasma model. Radiation transport was modeled using implicit Monte Carlo, and the simulations were started from room temperature. Tabulated equations of state are used for the helium and plastic window. In experiments, the window is deformed into a bubble-like shape due to the gas pressure. These simulations use a roughly spherical shape with a height of 600 μm and a thickness of 1.8 μm. A unique aspect of the HYDRA simulations is that they were run using a physics package that solves the resistive magneto-hydrodynamics (MHD) equations, including anisotropic electron thermal conduction due to the applied magnetic field, which is important for modeling these experiments. A form of generalized Ohm's law is also used35 which includes advection of the magnetic field due to temperature gradients, known as the Nernst term.
An example of the simulated electron temperature and density profiles, with and without an applied magnetic field, is shown in Figs. 11 and 12. These are 2D cross sections through the center of the target at the time when the laser heating pulse ends. The laser is incident on the target in vacuum from the left in this figure, and the initial position of the target window is at an axial position of 0 mm. A mixture of window material and helium extends from 0 to ∼2 mm in axial position. The blast caused by the window explosion is initially spherical and propagates into helium which causes some of the mixing. The heated helium gas region begins at an axial position of approximately 2 mm and extends to the right in these figures.
The simulations show an increase in the axial extent of laser heating with an applied magnetic field and a narrowing of the radial extent of heated material. Figure 13 shows a detailed comparison of the change in the radial profiles of the electron temperature and density with the applied magnetic field. This is consistent with the interpretation of the experimental results described in earlier sections. Higher peak temperatures are also present in the simulations with the axial magnetic field because of reduced electron thermal conductivity with an applied field. The electron Hall parameter ranges from approximately 1–5, which reduces the perpendicular conduction by factors of 0.1–0.01 relative to the parallel conductivity.
Extended propagation of the laser results from a combination of these magnetized effects leading to some thermal self-focusing of the beam. As the plasma temperature rises, it becomes more transparent to inverse bremsstrahlung absorption causing the laser to propagate further. The gas density also drops due to radial hydrodynamic motion which also increases the laser propagation depth. The hydrodynamic motion caused by the thermal pressure gradients creates a density profile that refracts the incident laser and focuses it in the gas. In the 3D simulations shown here, the laser does not continue to focus indefinitely but eventually breaks into filaments that manifest in the density variations seen in Fig. 12. This has also been observed in other 3D magnetized simulations of deuterium. Pondermotive effects in these simulations are generally small, but intensification of the beam by the thermally produced channels can lead to regions where ponderomotive thresholds are exceeded and so slightly increase propagation of the laser. Focusing in these channels can be limited by diffraction and backscatter, not accounted for in our HYDRA model, and so they warrant examination in the future with a more sophisticated laser-plasma code.
Synthetic x-ray images were calculated for a more direct comparison with the experimental x-ray pinhole camera images using the Spect3D software package from Prism Computational Sciences.23 SPECT3D is a multi-dimensional collisional-radiative spectral analysis code designed to simulate the atomic and radiative properties of laboratory and astrophysical plasmas. It used the HYDRA simulations of time-dependent electron temperature, density, and plasma composition to calculate the spectral properties and volume emission of the helium plasma with neon dopant at 0.25 ns intervals. An example x-ray emission calculation is shown in Fig. 14. For this calculation, the HYDRA simulation results were integrated over a time interval of 2 ns centered at the end of the laser heating pulse to replicate the time response of the detector used in the experiments. The x-ray transmission of a 6-micron-thick aluminum filter was also included in the x-ray emission calculation. Note that in this figure the displayed intensity was allowed to saturate in the plasma region with the mixed window material in order to more clearly show the x-ray emission in the helium-neon gas region. The x-ray emission intensity in the initial 2 mm region behind the initial target window location is calculated to be about 2× brighter than displayed in Fig. 14.
The calculated x-ray emission shows the same general spatial features as the calculated electron temperature distributions: it is longer and narrower with the applied magnetic field. This is consistent with the experimental measurements. The x-ray emission is lower, however, in the hottest central region of the gas for the case with magnetic field compared to without a magnetic field. This is due to a combination of lower central density and more fully ionizing the neon dopant at the calculated hottest temperatures. Neon was chosen for these experiments to provide a good diagnostic of the radial extent of the heated gas with the understanding that it is of limited use in determining the highest electron temperatures created.
VII. DISCUSSION AND CONCLUSIONS
Our experiments applied a magnetic field of 12 T to IB-heated plasmas and observed results in 2D transverse x-ray image sequences. We observed for the first time that this increases the plasma axial length scales by about 15% when the external magnetic field is applied. The magnetized laser-heated plasma also decreases in width by 10% with an applied magnetic field. These changes result from reduced radial thermal transport in a magnetic field, which has three important consequences: decreased IB heating coupling, enhanced hydrodynamic motion, and thermal self-focusing. Our experimental results are consistent with all of these effects taking place although deconvolving their relative importance will require further work. Prior experimental work which resolved only one spatial dimension or one moment of time may have had difficulty distinguishing between these competing effects.
With reduced radial transport, previous work suggests that central temperatures are expected to be higher by a factor of two or more. This increases the penetration of the heating beam axially by reducing inverse bremsstrahlung coupling upstream. Indeed, initially, from x-ray frames taken 0–1 ns into heating, x-ray emission indicates temperatures are higher with an applied field as the beam continues to heat further into the cold gas. 3D HYDRA simulations also predict a central temperature increase from ∼500 to ∼800 eV with a field and a central density drop from 60% to about 20% of initial density at 2 ns. The density shift is observed to increase with an applied magnetic field based on x-ray lineout images, which show a significant radial profile shift indicative of annular emission from a blast wave. We expect IB coupling to be more efficient in higher-density shocks forming on the plasma periphery compared to the central regions, where IB coupling efficiency is reduced. This allows the central part of the beam to penetrate further. This anisotropic absorption is included in simulations via numerical MHD modeling. In simulations with the magnetized case, anisotropic absorption creates a narrow ∼0.5 mm heated channel above 300 eV which extends beyond z = 8 mm. Simulations predict that this hot downstream penetration feature will be visible in x-ray images by applying emissivity models to simulation results (Fig. 14) and that propagation depth increases by 100%. However, this is not observed in experimental images, nor are TSF filaments; length increases by only 10%–20%.
We analyzed radial emissivity profiles from x-ray PHC image sequences to fit an analytical model for temperature and density. A thermal wave temperature profile is consistent with data in most cases even for early times and upstream regions when a magnetic field is present. Our analysis gives strong indications that a magnetic field reduces the overall temperature dependence in the cross field diffusion coefficient . This is because a narrower overall temperature profile was required to reproduce data, using an inverted parabolic temperature profile with y = 1 in Eq. (4) rather than a thermal wave produced at B = 0 when χ∼Te5/2. Such a profile is an approximate analytic fit to temperature profiles from simulations, representing a regime in which χ∼Te. This validates that the field reduces radial electron thermal heat transport. Other transport modes become important and control the outer plasma boundary growth, such as radiation diffusion for photons of energy below the opacity temperature of ∼25 eV. Hydrodynamic expansion drives energy transport after about two nanoseconds. Advection also drives the magnetic field out of the volume which creates a radial dependence for χ. Including these more subtle effects requires detailed simulations, and although simulations produced a good match to radial emissivity profiles, they failed to accurately predict the downstream extent of the plasma.
One limitation of our x-ray image analysis approach using analytic fits is that there are multiple possible configurations of temperature and density profiles that can produce the observed orthogonally projected emission. To address this, our image analysis used axisymmetric analytical temperature and density radial profiles that were informed both by analytical models of diffusion and our 3D simulations, with only a few free parameters. However, particularly because of the loss of temperature scaling in emission beyond 400 eV, this approach cannot be used to place strong constraints on temperature and overall energy density, only lower bounds. Our results are not strong constraints or measurements of overall deposited energy, but are significant in defining the geometry of heated regions.
Data analysis indicates enhanced temperature and density gradients with more rapid hydrodynamic motion with a magnetic field. Radial line-outs from x-ray profiles become centrally flattened upstream after 2–3 ns, indicating that emission is reduced in the center of the plasma and enhanced in a peripheral annulus as a blast wave develops. This effect is enhanced with an applied magnetic field and is easily observed in x-ray images because emissivity scales linearly with electron density. Fitting imaged emission profiles requires accounting for hydrodynamic expansion in the upstream areas which had been heated for a longer time in both magnetized and unmagnetized cases. Magnetized cases required a factor of two lower central density, down to 20% of initial. This is consistent with increased hydrodynamic motion from greater thermal gradients within a B-field and is supported by 3D simulations from HYDRA.
Simulations and prior research by Watkins and Kingham indicate that both with and without an applied field, the beam interacts with hydrodynamic motion-driven density gradients to create thermal self-focusing. This creates off-axis filamentation and whole-beam self-focusing, and the filaments spray outward off-axis. In simulations, the filament is enhanced with a magnetic field. However, experimental evidence for this is inconclusive. Bright spots in emission are present in x-ray images downstream of 7 mm in magnetized shots. These bright spots are narrower than the initial heating beam, indicating modifications to the beam profile, for example, from whole-beam convergence. Filamentation appears in visible light image frames with an applied field. The visible light images of filaments in the magnetized case are consistent with 3-D HYDRA models of filamented beam spray. However, x-ray images also do not show filaments which are expected to be present as in Fig. 14. These filaments may not confine sufficient laser power to heat the plasma to x-ray emission temperature. Further study is needed to understand why such downstream filaments and narrow beams are not present or do not heat the plasma significantly enough to be visible in x-rays.
Our results provide an important test of simulations for MagLIF. We provide the first measurements sensitive to magnetized filamentation and a comparison to simulations, finding that whole-beam focusing and filamentation were not as pronounced as simulations indicated. Experiments showed a lack of extended downstream narrow heated channels and forking which is expected in simulations, as is clear in Fig. 14. This suggests that heat transport may take place over large length scales, smoothing out smaller scale variations in temperature, density, and small-scale modified beam profiles.
Summarizing major findings, we saw a propagation depth that is significantly lower and has a larger diameter and less filamentation than simulations predicted. We characterized the overall shape and length of the laser-heated plasma volume in the presence of a MagLIF-like magnetic field, providing a lower boundary for deposited energy and temperature. The overall shape and length of the laser-heated channel provides important information about the location of deposited energy, which is important for MagLIF preheating. These results are significant for the MagLIF program because they aid in improving the accuracy of simulations, which are the only valid tool to further improve the design and progress of the experiments. Observing less penetration depth and filamentation allows designs with increased laser power, improving the crucial pre-heat in MagLIF. Understanding and measuring TSF and transport-mediated beam modification is also of utmost importance for MagLIF designs because such effects limit efficient coupling of laser energy into the target gas and consequently limit plasma temperatures and neutron yield. Understanding magnetized LPI is also important for MagLIF because hot-spots and filamentation reduce the efficiency of laser energy coupling to the preheated gas. Furthermore, filaments and downstream forking heated channels may impact target walls, causing undesirable ablation and contamination, which would lower the stagnation temperature and yield.
This work has limitations which future work may follow up on. One of the main avenues for improvements would simply be to collect more magnetized shot data with different conditions. While we have seen strong evidence for reduced transport, enhanced hydrodynamics, and increased TSF with a magnetic field, other diagnostics would be able to quantify the plasma density more clearly to definitively validate simulations. It would require higher laser frequencies than what we had available to penetrate the plasma for interferometry, but it could be framed on the 100 ps timescale and would provide definitive measurements of hydrodynamics and plasma refractive index to understand beam transport. Another limitation is the difficulty in diagnosing temperature because of the neon k-shell becoming fully ionized. Higher-Z dopant gasses or the use of bremsstrahlung emission from pure helium would enable direct photometry of x-ray emission. This can diagnose emissivity radial and axial profiles in a frame sequence at 2 ns time resolution. Since x-ray emissivity also scales with density, such a diagnostic would need the density profiles informed by another diagnostic to strongly bound temperature profiles and energy content.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, under Award No. DE-SC001900 and by the Center for High Energy Density Sciences. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
We wish to acknowledge the extensive work of the staff at the Z-Beamlet laser facility and for the opportunity to carry out student experiments in conjunction with capability development and detector testing. We also acknowledge the funding for S. Lewis by the Pulsed Power Program at Sandia National Laboratories.
Conflict of Interest
The authors have no conflicts to disclose.
The data that support the findings of this study are available from Sandia National Laboratories. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from the authors upon reasonable request and with the permission of Sandia National Laboratories.